Unveiling nontrivial fusion rule of Majorana zero mode using a fermionic mode

Fusing Majorana zero modes leads to multiple outcomes, a property being unique to non-Abelian anyons. Successful demonstration of this nontrivial fusion rule could serve as a smoking-gun evidence for its existence in topological superconducting systems. Here we show that this can be done by simply attaching a fermionic mode to a single Majorana zero mode. Our scheme requires periodic modulations of the energy level of this fermionic mode as well as its coupling with the Majorana mode. Adjusting the two quantities in different sequences would correspond to fusions of Majorana modes in different orders, ending up with distinct charge transfer process, which is detectable through charge sensing method or current measurement. Moreover, the charge transfer would experience an abrupt change with the same operations in the presence of trivial Andreev bound states. This provides a feasible way to distinguish Majorana modes from trivial Andreev bound states. In the end, we demonstrate our proposal in a nanowire-based topological superconductor coupled to a quantum dot, where the dot level and coupling strength are tunable in experiments.